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262                      ANALYTICAL MECHANICS
9.   A shell explodes at the highest point of its path and breaks up into two parts, the centers of mass of which lie in the line of motion.    Find the velocity of the pieces just after explosion, taking m for the mass of the shell, n for the ratio of the masses of the pieces, v for the velocity of the shell just before explosion, and E for the energy imparted to the pieces by the explosion.
10.  A particle slides down a smooth inclined plane which is itself free to move on a smooth horizontal plane.   Discuss the motions of the particle and of the plane.
11.  After falling freely through a height h a particle of mass m begins to pull up a greater mass M, by means of a string which passes over a smooth pulley.   Find the distance through which it will lift M.
12.   A smooth inclined plane which is free to move on a smooth horizontal plane is so moved that a particle placed on the inclined plane remains at rest.   Discuss the motion of the plane.
13.   A disk and a hoop slide along a smooth horizontal plane with the same velocity v, then begin to roll up the same rough inclined plane. How high will each rise?
14.   A ball, moving with a velocity v, collides directly with a ball at rest. The second ball in its turn collides with a third ball at rest.    If the masses of the first and last ball are mi and m3, respectively, show that the velocity acquired by the third ball is greatest when the mass of the second bail
satisfies the relation m% = VmiM3.
15.   Find the maximum velocity acquired by the third ball of the preceding problem.
16.   A billiard ball, moving at right angles to a cushion, impinges directly on an equal ball at rest at a distance d from the cushion.   Show
o //-that they will meet again at a distance rV~; d from the cushion.
17.   A ball is projected from the middle point of one side of a billiard table, so that it strikes an adjacent side first, then the middle of the opposite side.   Show that if I is the length of the adjacent side, the ball strikes the
adjacent side at a point  from the corner it makes with the opposite
18.   A simple pendulum hanging vertically has its bob in contact with a vertical wall.   The bob is pulled away from the wall and then it is let go. If e is the coefficient of restitution find the time it will take the pendulum to come to rest.
19.   A particle strikes a smooth horizontal plane with a velocity t;,